Explain what it means for a random process to be a Gaussian random process (GRP). Identify two functions (of time(s)) that characterize such a process. Suppose a GRP has the property of wide-sense stationarity (WSS) does it follow that it is also strict sense stationary (SSS)? Give an example of a process that is WSS but not SSS.

Show that the Brownian motion (BM) or (synonymously) Wiener process is Gaussian, and construct its joint probability distribution for an arbitrary sequence of measurement times. Is the process stationary? Calculate the product expectation [12 ] for a pair of times 1,2; is this quantity always positive? Suppose instead that a random walk process is defined in discrete time steps with a step taken in the positive / negative direction with equal probability in time , subject to the scaling relation () 2 = . Then consider the limit as 0 of this discrete process what is the relation between the limiting process and the continuous Brownian motion process as it is defined?